Riemann calculus is really needed only to describe curvature, not space-time.

The mathematical structure of space-time, like the mathematical structure of just space, is that of a manifold. A manifold is built on top of some other math concepts, I'm not sure if I should go into that level of detail, for the moment I won't.

We had another thread recently on why we combine space and time into a unified "space-time". This basically comes not from mathematics, but rather from special relativity.